Exact boundary controllability of unsteady flow in a tree-like network of open canals

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tree-like network of open canals

Ta-Tsien Li, Bopeng Rao

To cite this version:

Ta-Tsien Li, Bopeng Rao. Exact boundary controllability of unsteady flow in a tree-like network of open canals. 2005. �hal-00129710�

in a tree-like network of open canals

LI Ta-Tsien

Department of Mathematics, Fudan University Shanghai 200433, China

RAO Bopeng

Institut de Recherche Math´ematique Avanc´ee

Universit´e Louis Pasteur de Strasbourg, 67084 Strasbourg, France

Abstract : In this paper we establish the exact boundary controllability of unsteady flows in a tree-like network of open canals with general topology.

Key words : Exact boundary controllability, Saint-Venant system, tree-like network of open canals.

AMS Subject Classification :93B05, 35B37, 93C20, 35L50.

§1. Introduction.

The goal of this paper is to establish the exact boundary controllability of unsteady flows in a tree-like network of open canals.

The one-dimensional mathematical model of unsteady flows in an open canal was given by Saint-Venant system [1], which has been frequently used by hydraulic engineers in their practice (see also [2]-[7]). Using the theory on the semi-global C¹ solution and the exact boundary controllability for quasilinear hyperbolic systems (cf [10-15]), the exact boundary controllability of unsteady flows in both a single open canal and a star-like network of open canals was obtained in [8]. Later on, the exact boundary controllability of unsteady flows in a string-like network of open canals was established in [9].

In this paper we will generalize the previous results to a tree-like network of open canals with general topology. For this purpose, we will serve the total flux at the multiple node as an interface control. By means of a method different from that in [8], we establish another type of result on the exact boundary controllability of unsteady flows in a star-like network of open canals. Based on this new result and the results obtained in [8-9] on the exact boundary controllability of unsteady flows for a single open canal, a star-like network and a string-like network of open canals, we establish

the exact boundary controllability of unsteady flows in a tree-like network of open canals with general topology.

This paper is organized as follows. In §2 we give the statement of the exact boundary controllability of unsteady flows in a network of open canals and show the main idea of realizing it. In §3, we recall some results given in [8-9] on the exact boundary controllability of unsteady flows in a network of open canals with simpler topology. In §4, a fundamental Lemma on the continuation of piecewise C¹ solution in a star-like network of open canals is established. Then, combining the results in

§§3-4, we establish in §5 the exact boundary controllability of unsteady flows in a general tree-like network of open canals. Finally, some remarks are given in §6.

§2 Statement on the exact boundary controllability of unsteady flows in a network of open canals.

We consider unsteady flows in a network composed of N open canals. Let a_i0 and a_i1 be the x-coordinates of two ends of the i-canal and L_i = a_i1−a_i0 its length.

Suppose that there is no friction and the canal is horizontal and cylindrical, the corresponding Saint-Venant system can be written as (cf. [2], [8]-[9])

(2.1)











∂A_i

∂t + ∂(AiV_i)

∂x = 0,

t ≥0, a_i0 ≤x≤a_i1 (i= 1,· · ·, N),

∂V_i

∂t + ∂S_i

∂x = 0,

where, for the i-th canal, A_i = A_i(t, x) stands for the area of the cross section at x occupied by the water at time t, Vi = Vi(t, x) the average velocity over the cross section and

(2.2) Si = 1

2V_i²+ghi(Ai) +gYib,

where gis the gravity constant, constantYib denotes the altitude of the bed and

(2.3) hi =hi(Ai)

is the depth of the water, hi(Ai) being a suitably smooth function ofAi such that

(2.4) h⁰_i(Ai)>0.

The initial condition is

(2.5) t= 0 : (Ai, Vi) = (A_i0(x), V_i0(x)), a_i0 ≤x ≤a_i1 (i= 1,· · ·, N).

When a_i0 (resp. a_i1)is a simple node, we have the flux boundary condition : (2.6) x=a_i0 : AiVi =q_i0(t) (resp. x=a_i1 : AiVi =q_i1(t)).

While, when a_i0 (resp.a_i1) is a multiple node, we denote byJ_i0 (resp.J_i1) the set of indices corresponding to all the canals jointed ata_i0 (resp. a_i1). At a_i0 (resp. a_i1) we have (cf. [2],[8])the energy-type interface conditions

(2.7) Si =Sj, ∀j ∈ J_i0 (resp. ∀j ∈ J_i1) and the total flux interface condition

(2.8) X

j∈Jⁱ⁰

±AjV_j =Q_i0(t)

resp. X

j∈Jⁱ¹

±AjV_j =Q_i1(t) .

An equilibrium state (Ai, V_i) = (Ai, V_i) with A_i >0 (i= 1,· · ·N) is subcritical if

(2.9)











λ_i0 =Vi− q

gAih⁰_i(Ai)<0,

(i= 1,· · ·, N).

µ_i0 =V_i+ q

gA_ih⁰_i(Ai)>0,

For the purpose of the exact boundary controllability of unsteady flows in a network of open canals, we want to find a time T > 0 such that for any given initial data (Ai, Vi) = (A_i0(x), V_i0(x)) and final data (Ai, Vi) = (AiT(x), ViT(x)) (i= 1,· · ·N) in a C¹ neighbourhood of a given subcritical equilibrium state (Ai, V_i) (i = 1,· · ·N), there exist suitable flux boundary controls at some simple nodes and suitable total flux interface controls at some multiples nodes, such that the corresponding mixed initial-boundary value problem for Saint-Venant system (2.1) with the initial data (2.5), the corresponding boundary conditions (2.6) and the interface conditions (2.7)- (2.8) admits a unique piecewise C¹ solution on the time interval 0 ≤ t ≤ T, which satisfies exactly the final condition

(2.10) t=T : (Ai, Vi) = (AiT(x), ViT(x)), a_i0 ≤x≤a_i1 (i= 1,· · ·, N).

To this end, it suffices to construct a piecewiseC¹ solution to Saint-Venant system (2.1) on the time interval 0≤t ≤T, which

satisfies the initial condition (2.5), the final condition (2.10), all the energy-type interface conditions (2.7) and a part of flux boundary conditions (2.6) and total flux interface conditions (2.8). Once this solution is obtained, we put it into the rest of flux boundary conditions (2.6) and total flux interface conditions (2.8) to get the

desired flux boundary controls q_i0(t) (resp.q_i1(t)) and total flux interface controls Q_i0(t) (resp. Q_i1(t)), which realize the exact boundary controllability for the Saint- Venant system.

To construct this kind of piecewise C¹ solution is a nonstandard problem which can be regarded as an inverse problem. There is no uniqueness in general. However we can get this solution in a stable way, i.e., by solving several well-posed problems.

The main principle in the resolution consists in making the number of controls or the controllability timeT as small as possible. However, for a smaller number of controls, we need a larger controllability time, and vise versa.

§3 Construction of piecewise C¹ solution in a network of canals with simpler topology.

In this section, we recall some results given in [8-9], which will be useful in what follows.

1). Consider a single canal with flux boundary conditions at one end. If

(3.1) T >2Lmaxn

− 1 λ₀, 1

µ₀ o

,

whereλ₀ andµ₀ are given by (2.9) andL=a₁−a₀ is the length of the canal, we can construct aC¹ solution on the single canal, such that it satisfies any given initial and final conditions in aC¹ neighbourhood of a given subcritical equilibrium state(A, V) as well as the the flux boundary condition on one end (see [8] for the detail). We illustrate this procedure by the following figure :

Figure 3.1. A single canal

Here a bold point ”•” denotes the starting end on which the flux boundary condition is satisfied, while a hollow point ”◦” denotes the arriving end, on which the value of solution is determined by the procedure of resolution, and the arrow denotes the direction along which we solve the problem.

2). By rescaling and folding all correspondingx-interval to the interval 0≤x ≤1, the previous method can be applied to a star-like network of N open canals (see [8]

for the detail). Let T satisfy

(3.2) T >2 max

1≤i≤N

Limaxn

− 1 λ_i0, 1

µ_i0 o

,

whereλ_i0 andµ_i0 are given by (2.9) and L_i =a_i1−a_i0 is the length of thei-th canal (i= 1,· · ·N)). Then, by solving the problem from the joint end to all simple nodes, we can get a piecewiseC¹ solution which satisfies any given initial and final conditions in

a piecewiseC¹ neighbourhood of a given subcritical equilibrium (Ai, V_i) (i= 1,· · ·N) as well as the interface conditions (2.7)-(2.8) at the joint end (the multiple node). We illustrate this procedure by the following figure :

Figure 3.2. A star-like network of canals

Here all simple nodes are the arriving ends ”◦” on which the solution is determined by the procedure of resolution. The joint node is the starting end ”•” on which all the interface conditions (2.7)-(2.8) are satisfied. The procedure of resolution is indicated by the arrows : from ”•” to ”◦”.

3). For a string-like network of N open canals, if

(3.4) T >2

N

X

i=1

Limaxn

− 1 λ_i0, 1

µ_i0 o

,

where λ_i0 and µ_i0 are given by (2.9) and Li = a_i1−a_i0 is the length of i-th canal (i = 1,· · ·N)), by solving the problem from one simple node ”•” to another simple node ”◦”, indicated by Figure 3.3, we can still get a piecewise C¹ solution which satisfies any given initial and final conditions in a piecewise C¹ neighbourhood of a given subcritical equilibrium (A_i, V_i) (i = 1,· · ·N) as well as the flux boundary condition on one simple node and the interface conditions (2.7)-(2.8) on all the multiple nodes denoted by ” ” (see [9] for the detail). This situation is similar to the case shown by Figure 3.1 for a single canal, however the controllability time should be much larger.

Figure 3.3. A string-like network of canals

4). Based on the results given in 2) and 3), if T is large enough (bigger than the maximum of the exact controllability time corresponding to each string-like sub-network), similarly we can get a piecewise C¹solution for a star-like network of string-like sub-networks, which satisfies any given initial and final conditions in a piecewiseC¹ neighbourhood of a given subcritical equilibrium as well as the interface conditions (2.7)-(2.8) on each multiple node. We illustrate this procedure by the following figure :

Figure 3.4. A star-like network of string-like sub-networks

Here the joint node of all the string-like sub-networks is the starting end ”•”, all the simple nodes are the arriving end ”◦”, while all the joint nodes in the string-like sub-networks are denoted by ” ” on which the interface conditions (2.7)-(2.8)are satisfied.

§4. Fundamental Lemma on the continuation of piecewiseC¹ solution.

In order to get a desired piecewise C¹ solution for a network of N open canals with general topology, we should extend the piecewise C¹ solution on a network of open canals with simpler topology shown in the previous section. To this end, we now give another way to construct a piecewise C¹ solution for a star-like network of N open canals by assuming that a C¹ solution satisfying any given initial and final conditions in a C¹ neighbourhood of a corresponding subcritical equilibrium is obtained on one canal, and then solving the problem from the joint end to other simple ends respectively, which is illustrated in the following figure :

Figure 4.1 A star-like network of canals

Let (A₁, V₁) be a known C¹ solution on the first canal denoted by ” ”. In order to construct the desired C¹ solution on other canals in the star-like network, we serve the energy-like interface condition as a boundary condition at the joint node x= 0 for each corresponding canal. Thus the piecewise C¹ solution constructed satisfies only the energy-like interface conditions (2.7), while the total flux interface condition (2.8) is no longer satisfied at the joint node which is then denoted by ” ”.

More precisely, we have the following result.

Lemma 4.1Let (Ai, V_i) = (Ai, V_i) (i= 1,· · ·, N) be a subcritical equilibrium state.

Suppose that

(4.1) T >2 max

i=2,···,N

n

L_imax

− 1 λ_i0, 1

µ_i0 o

,

where λ_i0 and µ_i0(i = 2,· · ·, N) are still given by (2.9). Let (Ai0(x), Vi0(x)) and (AiT(x), ViT(x)) (i= 1,· · ·, N) be any given initial state and final state with smallC¹ norms

n

P

i=1

k(Ai0(·)−Ai, V_i0(·)−Vi)kC¹[0,Lⁱ]and

n

P

i=1

k(AiT(·)−Ai, V_iT(·)−Vi)kC¹[0,Lⁱ], which satisfy the conditions of piecewise C¹ compatibility with the energy-like interface conditions (2.7) at the points (t, x) = (0,0) and (T,0) respectively.

Let

(4.2) Ri(T) ={(t, x)|0≤t≤T, 0≤x≤Li} (i = 1,· · ·, N).

Assume that we have already determined the solution (A1, V₁) with small C¹ norm k(A₁−A₁, V₁−V₁)k_C¹_[R1(T)], which satisfies the initial condition (2.5) and the final condition (2.10). Then there exists a piecewise C¹ solution (Ai, Vi)(i = 1,· · ·, N) on the domain∪^N_i=1R_i(T) with small piecewiseC¹norm

n

P

i=1

k(Ai−A_i, V_i−Vi)kC¹[Rⁱ(T)], which satisfies the initial condition (2.5), the final condition (2.10) and the energy-like interface conditions (2.7) at the joint node x= 0.

Proof.Fori= 1,· · ·, N, introducing the Riemann invariantsr_i ands_i as follows : (4.3)

2ri =Vi−Vi−Gi(Ai), 2s_i =V_i−V_i+G_i(A_i), where

(4.4) G_i(A_i) =

Z Aⁱ Ai

s

gh⁰_i(Ai) A_i dA_i.

Let H_i be the inverse function ofG_i(Ai). We have (4.5)

V_i =r_i+s_i+V_i, A_i =H_i(si−r_i) and

(4.6) H_i⁰(si−r_i) =

s A_i

gh⁰_i(Ai) >0.

Since (Ai, V_i) = (Ai, V_i) correspond to (ri, s_i) = (0,0), we get

(4.7) Hi(0) =Ai, H_i⁰(0) = s

Ai

gh⁰_i(Ai) >0.

Taking (ri, s_i) (i = 1,· · ·, N) as new unknown variables, system (2.1) can be reduced to the following system of diagonal form :

(4.8)











∂r_i

∂t +λ_i∂r_i

∂x = 0,

(i= 1,· · ·N),

∂si

∂t +µ_i∂si

∂x = 0, where

(4.9)







λi =Vi−p

gAih⁰_i(Ai)<0,

(i= 1,· · ·N).

µ_i =V_i+p

gA_ih⁰_i(Ai)>0,

The energy-type interface conditions (2.7) now become x= 0 : 1

2(ri+si+Vi)²+ghi(Hi(si−ri)) +gybi

(4.10)

=1

2(r1+s₁+V_i)²+gh₁(H1(s1 −r₁)) +gy_b1 (i= 2,· · ·N).

For each i= 2,· · ·, N, let P_i(ri, s_i, r₁, s₁) =1

2(ri+s_i+V_i)² +gh_i(Hi(si−r_i)) +gy_bi (4.11)

−1

2(r₁+s₁+V₁)²+gh_i(H₁(s₁−r₁)) +gy_b1 .

Since the state (Ak, V_k) = (Ak, V_k) corresponds to (rk, s_k) = (0,0) for k = 1,· · ·N, condition (4.10) implies

(4.12) P_i(0,0,0,0) = 0.

On the other hand, a straightforward computation gives

(4.13) ∂P_i

∂s_i(0,0,0,0) =V_i+ q

gA_ih_i(Ai)>0.

Then, by the implicit function theorem, we can find a C¹ function f_i in a neighbourhood of (ri, r₁, s₁) = (0,0,0) such that (4.10) can be equivalently rewritten as

(4.14) si =fi(ri, r₁, s₁)

with f_i(0,0,0) = 0. Since the solution (r1, s₁) is known, putting the values r₁(t,0) ands₁(t,0) into (4.14) and notingfi(0,0,0) = 0, (4.14) can be equivalently rewritten as

(4.15) si =Gi(t, ri) +Hi(t), where G_i andH_i are C¹ functions such that

(4.16) G_i(t,0)≡0

and

(4.17) kHik_C¹_[0,T] is small enough.

Similarly, for eachi = 2,· · ·, N, we have

(4.18) ∂Pi

∂ri

(0,0,0,0) =Vi− q

gAihi(Ai)<0.

Then, in a neighbourhood of (ri, si, r₁, s₁) = (0,0,0,0), condition (4.10) can be equivalently rewritten as

(4.19) r_i=G_i(t, si) +H_i(t), where Gi andHi areC¹ functions such that

(4.20) Gi(t,0)≡0

and

(4.21) kHik_C¹_[0,T] is small enough.

Moreover, under the assumption of Lemma 4.1, it is easy to see that the conditions ofC¹compatibility with (4.15)(resp.(4.19)) are still satisfied at the point (t, x) = (0,0) (resp. (T,0)). Then, applying the results given in [13] and [8] to system (4.8) for the i-th canal, we get the solution (ri, s_i), i.e., (Ai, V_i) for each i = 2,· · ·, N, which, together with (A₁, V₁), satisfy the initial condition (2.5), the final condition (2.10) and the energy-type interface conditions (2.7) atx = 0. The proof is then complete.

Remark 4.1 Lemma 4.1 is still valid for a star-like network of string-like sub- networks, when a piecewise C¹ solution is known on one string-like sub-network and T >0 is suitably large.

Figure 4.2. A star-like network of string-like sub-networks

§5. Exact boundary controllability of unsteady flows in a tree-like network of open canals.

A network is called to have atree-likeconfiguration, if any two nodes in the network can be connected by a unique path of canals. In other words, a tree-like network is a connected network without loop. In what follows, for T >0 suitably large, taking the results given in§3 as the first step, we will successively use Lemma 4.1 or Remark 4.1 to get a desired piecewise C¹ solution on the whole network, which implies the exact boundary controllability for a tree-like network of open canals.

Arbitrarily choosing a node as the starting end, we construct a desired piecewise C¹ solution as follows.

Case A. When the starting end Ok is a simple node, on which the boundary flux qk with smallC¹ normkqk(·)−AkVkk_C¹_[0,T] is given, we assume that the initial data and the final data satisfy the conditions of C¹ compatibility with the flux boundary condition (2.6) atO_k. ForT >0 suitably large, by the procedure illustrated in Figure 3.1, we can get a C¹ solution of system (2.1) on the canal with the end Ok (another end of this canal is denoted byO_l), which satisfies the corresponding initial condition (2.5) and the final condition (2.10) on the canal and the flux boundary condition (2.6) at the simple node Ok. This procedure can be shown in the following figure :

Ok Ol

Figure 5.1. Starting from a simple node

Still for T > 0 suitably large, applying Lemma 4.1 illustrated in Figure 4.1 to the local star-like network of open canals jointed at the multiple node O_l, we get a piecewiseC¹ solution which satisfies the corresponding initial condition (2.5) and the final condition (2.10) on this local star-like network, the flux boundary condition (2.6)

at the simple node O_k, and the energy-like interface conditions (2.7) at the multiple node Ol. We illustrate this procedure by the following figure :

O_k O_l

Figure 5.2. Extension to a local star-like network

Then, repeating the procedure given by Lemma 4.1, we can construct a piecewise C¹ solution on the whole network of canals, which satisfies the initial condition (2.5) and the final condition (2.10) on the whole network, the boundary condition (2.6) at the starting end O_k and the energy-like interface conditions (2.7) at all multiple nodes. Finally, we substitute this solution in (2.8) to get the total flux interface controls Q_i(t) for all multiple nodes, and in (2.6) to get the flux boundary controls q_i(t) for all simple nodes except the starting endO_k. We illustrate this procedure by the following figure :

Ok

starting end Ol

Figure 5.3. Exact controllability starting from a simple node (23 controls for 23 canals)

Case B. When the starting end O_l is a multiple node, on which the total flux Ql with small norm kQl(·)−P

j∈Jl±AjVjk_C¹_[0,T] is given, we assume furthermore that the initial data and the final data also satisfy the conditions of piecewise C¹ compatibility with the total flux interface condition (2.8) at the multiple nodeO_l. For T >0 suitably large, by the procedure illustrated in Figure 3.2, we get a piecewiseC¹ solution for the local star-like network jointed at the multiple nodeOl, which satisfies

the corresponding initial condition (2.5) and the final condition (2.10) on the whole local star-like network, the interface conditions (2.7) and (2.8) at multiple node Ol. We illustrate this procedure by the following figure.

O_l

Figure 5.4. Starting from a multiple node

Then repeating the procedure given by Lemma 4.1 as shown in the previous situation, we finally get a piecewise C¹ solution for the whole network of canals, which satisfies the initial condition (2.5) and the final condition (2.10) on the whole network, the total flux interface condition (2.8) at the starting multiple node Ol and the energy-type interface conditions (2.7) at all multiple nodes. Thus, in a similar way, we can define the flux boundary controlsq_i(t) for all simple nodes, and the total flux controls Qi(t) for all multiple nodes except the starting end Ol, respectively. We illustrate this procedure by the following figure :

Ol

starting end

Figure 5.5. Exact controllability starting from a multiple node (23 controls for 23 canals)

Case C. If the tree-like network contains string-like sub-networks, we can use the procedure shown in Figure 3.3 or Figure 3.4

as the first step and then apply repeatedly Remark 4.1 to get a desiredC¹piecewise solution for the whole network of the canals (see Figures 5.6 and 5.7). In this way we can reduce the number of controls, however, the controllability time should be correspondingly enlarged (cf. (3.4)).

starting end

Figure 5.6. Exact controllability starting from a simple node

for a tree-like network containing string-like sub-networks (15 controls for 23 canals)

starting end

Figure 5.7. Exact controllability starting from a multiple node

for a tree-like network containing string-like sub-networks (15 controls for 23 canals)

§6 Some remarks

1). The exact controllability time obtained in each case is optimal.

2). The flux boundary condition (2.6) can be replaced by the water level boundary condition

(6.1) x=a_i0 : Ai =q_i0(t).

Accordingly, the energy-like interface conditions (2.7) can be also replaced by the water level interface conditions

(6.2) hi(Ai) +Yib =hj(Aj) +Yjb, ∀j ∈ J_i0.

3). The flux boundary condition (2.6) can be replaced by the following boundary condition of underflow gate (cf.[7])

(6.3) x=a_i0 : A_iV_i|AiV_i|= 2gu²_i0(y_i0−h_i(Ai)),

wherey_i0is the water level outside thei-th canal andu_i0 is the height of gate opening at x=a_i0.

Accordingly, on each intermediate joint end a_i0 of a string-like sub-network, the interface conditions (2.7)-(2.8) can be also replaced by the interface conditions of underflow gate (cf.[7])

(6.3) A_i−1V_i−1|A_i−1V_i−1|=AiVi|AiVi|= 2gu²_i0(h_i−1(A_i−1)−hi(Ai)).

4). When one considers the friction in a network of open canals, Saint-Venant system (2.1) becomes

(6.4)











∂Ai

∂t + ∂(AiVi)

∂x = 0,

t ≥0, a_i0 ≤x ≤a_i1 (i = 1,· · ·, N)

∂V_i

∂t + ∂S_i

∂x +F_i(Ai, V_i) = 0

with Fi(Ai,0) ≡ 0 and ViFi(Ai, Vi) ≥ 0. For any given equilibrium state (Ai, V_i) = (Ai, V_i) with A_i > 0(i = 1,· · ·, N), we still have the exact boundary controllability.

5). Since both the initial and final states belong to a piecewiseC¹ neighbourhood of an arbitrarily given subcritical state, our results give only the local exact boundary controllability for unsteady flows in a tree-like network of open canals, however, that is enough to get the corresponding global exact boundary controllability. In fact, in order to drive any initial state given in a piecewise C¹ neighbourhood of a subcritical state to any final state given in a piecewise C¹ neighbourhood of another subcritical state, it suffices to use the previous results to drive the initial state to the first equilibrium state, then successively transfer the first equilibrium state to the second equilibrium state, and finally reach the final state. Thus we get the corresponding global exact boundary controllability, however, the controllability time must be enlarged many times.

Acknowledgments. LI Ta-Tsien is supported by the Special Funds for Major State Basic Research Projets of China. The paper was written during Rao Bopeng’s visit in the Institut Sino-Fran¸cais de Math´ematiques Appliqu´ees (ISFMA) at Fudan University on 2003- 2004.

Bibliography

[1] de Saint-Venant, B., Th´eorie du mouvement non permanent des eaux, avec application aux crues des rivi`eres et l’introduction des mar´ees dans leur lit, C.R.Acad.Sci., 73(1871), 147-154, 237-240.

[2] Leugering, G. and Schmidt, E.G., On the modelling and stabilization of flows in networks of open canals, SIAM J. Control Optim., 41(2002), 164-180.

[3] Gugat, M. and Leugering, G., Global boundary controllability of the de St.Venant equations between steady states, Ann. I. H. Poincar´e—AN20, 1(2003), 1-11.

[4] Coron,J.-M., d’Andr´ea-Novel, B. and Bastin, G.,A Lyapunov approach to control irrigation canals modeled by Saint-Venant equations, in CD-Rom Proceedings, Paper F1008-5, ECC99, Karlsruhe, Germany, 1999.

[5] de Halleux, J., d’Andr´ea-Novel, B., Coron, J.-M. and Bastin, G., A Lyapunov approach for the control of multi reach channels modelled by Saint-Venant equations, in CD-Rom Proceedings, NOLCOS’01, St-Petersburg, Russia, June 2001, 1515- 1520.

[6] de Halleux, J. and Bastin, G., Stabilization of Saint-Venant equations using Riemann invariants : Application to waterways with mobile spillways, in CD-Rom Proceedings, Barcelona, Spain, July 2002.

[7] de Halleux, J., Prieur, C., Coron, J.-M., d’Andr´ea-Novel, B. and Bastin, G., Boundary feedback control in networks of open channels, Automatica 39 (2003), 1365-1376.

[8] Li T.T., Exact boundary controllability of unsteady flows in a network of open canals, to appear in Math. Nachr.

[9] Li, T.T. Exact boundary controllability for quasilinear hyperbolic systems and its application to unsteady flows in a network of open canals, Math. Meth. Appl. Sci., 27(2004), 1089-1114.

[10] Li, T.T. and Jin, Y., Semi-global C¹ solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems, Chin. Ann. Math., 22B (2001), 325-336.

[11] Li, T.T., Rao, B.P. and Jin, Y., Solution C¹ semi-globale et contrˆolabilit´e exacte fronti`ere de syst`emes hyperboliques quasi lin´eaires, C.R.Acad.Sci.Paris, t. 333, S´erie I (2001), 219-224.

[12] LIi, T.T. and Rao, B.P., Exact boundary controllability for quasilinear hyperbolic systems, SIAM J. Control Optim., 41(2003), 1748-1755.

[13] Li, T.T. and Rao, B.P.,Local exact boundary controllability for a class of quasilinear hyperbolic systems, Chin. Ann. Math., 23B(2002), 209-218.

[14] Li T.T. and Yu L.X. Exact controllability for first order quasilinear hyperbolic systems with zero eigenvalues, Chin. Ann. Math., 24B(2003), 415-422.

[15] Li T.T. and Yu W.C., Boundary Value Problems for Quasilinear Hyperbolic Systems, Duck University Mathematics Series V, 1985.